MARCH 24, 2026 -- A paradigmatic NMR experiment includes polarization of nuclear spins, commonly protons, followed by applying a radio frequency pulse, which induces detectable transverse magnetization. When the interaction between spins comes into play, such as dipole-dipole interaction in solids, not only the transitions of individual spins are possible, but also the concert transitions in groups of spins. These multiple quantum (MQ) transitions are forbidden in the strong magnetic field. Thus, KeAi cannot simply apply doubled radio frequency (energy) to flip two spins in concert.

Instead, MQ transitions are excited indirectly by a specially designed pulse sequence and then converted back to observable magnetization. The response of MQ coherences to the phases of the pulses depends on the order (number of quanta) of the transition, which is exploited to resolve the transitions of different order in MQ spectra in two-dimensional experiment.

MQ NMR spectroscopy is useful in investigation of geometry of liquid crystals, reducing the indiscernible transitions to manageable number in high-order spectra, determination of number of spins in finite clusters, which coincides with the highest order transition, in investigation of polymer networks, where decreased mobility near crosslinks gives discernable MQ signals, which is hardly accessible to other atomic level techniques. On the other hand, MQ NMR coherences are valuable as a model of interacting qubits for quantum computation.

Excitation of MQ coherences, however, takes time — in the course of which the decay of states also occurs. This process differs from common transverse NMR relaxation and less studied. To that end, KeAi investigated the simplest system consisting only of two spins, but this system is subjected to the interactions with surroundings, though much weaker. The theory was developed based on Lindblad equation describing the dynamics of open quantum systems. The action of the environment was imposed by the dephasing Lindblad operators, flipping each spin of the system independently.

The experimental validation requires a quite rare system — an ensemble of well-isolated spin pairs. Some hydrate crystals are apparent candidates, but close location of protons results in very intense dipolar coupling, deteriorating the performance of MQ experiments. KeAi find the system with larger spacing in hambergite crystal. The protons in hambergite constitute well-isolated planar zigzag chains. In this geometry, KeAi can adjust the direction of the field to form "magic angle", which eliminates the dipolar coupling, for odd or even pairs of spins in the chain, leaving well-isolated spin pairs weakly interacting with remote spins (as shown in the figure). These two orientations results in different dipolar couplings within spin pairs, as well as with the environment. The performed MQ experiments demonstrate agreement with the theory: the intensity oscillates between coherences of order 0 and 2 with increasing MQ excitation time, with frequency determined by the dipolar coupling in the pair; the overall intensity damp exponentially. The two orientations differ not simply by the timescale. In the case of 2MA the tail of the decay lose oscillatory character and intensities simply decay exponentially, though the relaxation time is longer compared to 1MA. In the case of orientation 2MA the interactions overly become weaker, both within the pairs and environment, compared to 1MA. The oscillations are consequence of the coherent dynamics within spin pairs, while the decay characterizes the strength of the interactions with the environment.

Though the appreciable MQ signal is detectable at long preparation times, the characteristic spin-pair behavior fades faster. Thus, the preservation of the coherent evolution in spin system requires both strong interactions within the system, as well minimization of its coupling with environment.